Planar nonautonomous polynomial equations V. The Abel equation

Wilczyński, Paweł

doi:10.7494/opmath.2013.33.1.175

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Algebraic Morphisms Of Quasi-lie Schemes and Abel Equations1

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2015

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“…This system of differential equations describes several cases of polynomial differential equations appearing in the XVI Hilbert's problem [35]. Moreover, the study of planar polynomial equations has also being attracting some attention [37,47]. By defining S(W 2 , V 2 ) with W 2 = Z 0 , Z 1 and V 2 = Z 0 , .…”

Section: Algebraic Morphisms Of Quasi-lie Schemes and Abel Equationsmentioning

confidence: 99%

Quasi-Lie families, schemes, invariants and their applications to Abel equations

Cariñena

Lucas

2015

Journal of Mathematical Analysis and Applications

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This work analyses types of group actions on families of t-dependent vector fields of a particular class, the hereby called quasi-Lie families. We devise methods to obtain the defined here quasi-Lie invariants, namely a kind of functions constant along the orbits of the above-mentioned actions. Our techniques lead to a deep geometrical understanding of quasi-Lie schemes and quasi-Lie systems giving rise to several new results. Our achievements are illustrated by studying Abel and Riccati equations. We retrieve the Liouville invariant and study other new quasi-Lie invariants of Abel equations. Several Abel equations with a superposition rule are described and we characterise Abel equations via quasi-Lie schemes.

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Section: Algebraic Morphisms Of Quasi-lie Schemes and Abel Equationsmentioning

confidence: 99%

Quasi-Lie families, schemes, invariants and their applications to Abel equations

Cariñena

Lucas

2015

Journal of Mathematical Analysis and Applications

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Planar nonautonomous polynomial equations V. The Abel equation

Abstract: Abstract. We give a full description of the dynamics of the Abel equationż = z 3 + f (t) for some special complex valued f . We also prove the existence of at least three periodic solutions for equations of the formż = z n + f (t) for odd n ≥ 5.

Cited by 1 publication

References 7 publications

Quasi-Lie families, schemes, invariants and their applications to Abel equations

Quasi-Lie families, schemes, invariants and their applications to Abel equations

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